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GATE 2024 MA – Question 45

Ordinary Differential Equations · Higher Order Equations and Systems · 2 marks · Multiple choice

A is a nondiagonalizable real 2×2 matrix with real eigenvalue λ and eigenvector v. Choose the general solution of y′=Ay, with arbitrary constants c₁,c₂ and (A−λI)u=v when u appears.

  1. $c_1e^{\lambda t}v+c_2te^{\lambda t}v$
  2. $c_1e^{\lambda t}v+c_2t^2e^{\lambda t}v$
  3. $c_1e^{\lambda t}v+c_2e^{\lambda t}(tv+u)$
  4. $c_1e^{\lambda t}v+c_2te^{\lambda t}(v+u)$

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Correct answer: (C) $c_1e^{\lambda t}v+c_2e^{\lambda t}(tv+u)$

Explanation

A single length-two Jordan chain satisfies Av=λv, Au=λu+v. Its exponential sends u to e^(λt)(u+tv). Together with e^(λt)v this gives two independent solutions. **C**.

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